论文标题
多项式的根和临界点 - 迪克森代数
Roots and Critical Points of Polynomials over Cayley--Dickson Algebras
论文作者
论文摘要
我们研究了在任意领域和任意维度上的Cayley-Dickson代数上多项式的根。为此，我们概括了从四元组和八元多项式到这种设置的球形根的概念，并展示了它们的基本特性。我们表明，多项式$ f（x）$的球形根（但并非所有根）也是其伴随多项式$ c_f（x）$的根（定义为$ f（x）$）。 For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$, and we also prove that all roots of $f'(x)$ are contained in the snail of $f(x)$, as defined by Ghiloni and Perotti for quaternions.后两个结果将经典的高斯定理推广到局部复杂的cayley-迪克森代数，我们还将詹森关于真实多项式的经典定理推广到此设置。
We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial $f(x)$ are also roots of its companion polynomial $C_f(x)$ (defined to be the norm of $f(x)$). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$, and we also prove that all roots of $f'(x)$ are contained in the snail of $f(x)$, as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.
